• 沒有找到結果。

# A tree in which all non-endvertices having distinct statuses is status unique in the family of all trees

## ઞᮽ䌘

### Conjugate 1: A tree in which all non-endvertices having distinct statuses is status unique in the family of all trees

A branch [6, 7, 8, 9] at a vertex b in a tree is a maximal subtree containing b as an endvertex. Let B1, B2, . . . , Bm(M ≥ 1) be several distinct branches at b in the tree. Assume that U is the subtree induced by V (B1)∪V (B2)∪·∪V (Bm).

Then U is called a union branch [5] at b. The branch-weight [6, 7, 8, 9] of b is the maximum number of edges in any branch at b. The branch-weight sequence [6, 7, 8] of a tree is the list of the branch-weights of all vertices arranged in non-increasing order. Let T be a tree and x, y be a pair of distinct non-endvertices in T with the same status. Suppose that A is a union branch at x and B is a union branch at y, where V (A) ∩ V (B) = ϕ and |V (A)| = |V (B)|. Let C be the subgraph of T induced by the vertex set (V (T ) − (V (A) ∪ V (B))) ∪ {x, y}. Let Tbe the tree constructed from A, B, and C by identifying the vertex x in A with the vertex y in C, and identifying the vertex y in B with the vertex x in C. It is shown that T and T have the same status sequence [4]. We call such a tree T a status-retained transfer of T. We now propose the following conjectures.

Conjugate 2: Assume that T0 is a tree and there is at least a pair of distinct non-endvertices of T0 with the same status. Let T be a tree. Then, T has the

### 53

same status sequence as T0 if and only if there exist trees T1, T2, . . . , Tk (k∈ N) such that Ti is a status-retained transfer of Ti−1 for i = 1, 2, . . . , k and Tk≃ T . Conjugate 3: Two trees have the same branch-weight sequence if they have the same status sequence.

Remark: Supported by the Ministry of Science and Technology of R.O.C. un-der grants MOST 105-2115-M-424-001, MOST 106-2115-M-424-001.

Keywords: status, status sequence, branch-weight, branch-weight sequence, tree.

### References

[1] F. Buckley, F. Harary, Unsolved problems on distance in graphs, Electron.

Notes Discrete Math. 11 (2002) 89-97.

[2] L. Pachter, Constructing status injective graphs, Discrete Appl. Math. 80 (1997) 107-113.

[3] J.-L. Shang, C. Lin, Spiders are status unique in trees, Discrete Math. 311 (2011) 785-791.

[4] J.-L. Shang, On constructing graphs with the same status sequence, Ars Comb. 113 (2014) 429-433.

[5] J.-L. Shang, T.-W. Shyu, C. Lin, Weakly status injective trees are status unique in trees, Ars Comb. 139 (2018) 133-143.

[6] J.-L. Shang, C. Lin, An uniqueness result for the branch-weight sequences of spiders, Whampoa-An Interdisciplinary Journal 54 (2008) 31-38.

[7] R. Skurnick, Extending the concept of branch-weight centroid number to the vertices of all connected graphs via the Slater number, Graph Theory Notes of New York 33 (1997) 28-32.

[8] R. Skurnick, A characterization of the centroid sequence of a caterpillar, Graph Theory Notes of New York 41 (2001) 7-13.

[9] P.J. Slater, Accretion centers: A generalization of branch weight centroids, Discrete Appl. Math. 3 (1981) 187-192.

### E-mail: h16ds202@hirosaki-u.ac.jp

The root of matrices is a classical problem in matrix theory which can be traced back to the work of Arthur Cayley in 1858. However, not much is known about the question of existence of entrywise nonnegative square roots for a nonnegative matrix. We will consider the nonnegative roots of rank-one matrices and circulant matrices, etc. The necessary and sufficient conditions for the existence of the nonnegative pthroot of a circulant matrix with the order 3 and 4 will be given. Moreover, it is proved that the roots of a circulant matrix are circulant if and only if its eigenvalues are all distinct.

Keywords: Circulant matrix, nonnegative matrix, matrix roots

### References

[1] A. Cayley, A memoir on the theory of matrices, Phil. Trans. Roy. Soc.

London, 148 (1858), 17-37.

[2] A. Cayley, On the extraction of the square root of a matrix of the third order, Proc. Roy. Soc. Edinburgh, 7 (1872), 675-682.

[3] N.J. Higham and L.-J. Lin, On pth roots of stochastic matrices, Linear Algebra Appl., 435 (2011), 448-463.

[4] P.-R. Huang, Nonnegative roots for circulant matrices of order less than four, submitted.

[5] R. Loewy and D. London, A note on an inverse problems for nonnegative matrices, Linear Multilinear Algebra, 6 (1978), 83-90.

[6] B.-S. Tam and P.-R. Huang, Nonnegative square roots of matrices, Linear Algebra Appl., 498 (2016), 404-440.

### E-mail: macws@math.sinica.edu.tw

Let n be a positive integer, q = 2n, and let Fq be the finite field with q elements. For each positive integer m, let Dm(X) be the Dickson polynomial of the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of q + 1. We study the existence of α∈ Fq such that Dm(α) = Dm−1) = 0. We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.

Keywords: absolutely irreducible, button madness, Dickson polynomials, Fermat number, finite field, reciprocal polynomial

### References

[1] A. Blokhuis and A. E. Brouwer, Button madness, available at http://www.win.tue.nl/∼aeb/preprints/madaart2c.pdf.

[2] W.-S. Chou, The factorization of Dickson polynomials over finite fields, Finite Fields Appl. 3 (1997), 84-96.

[3] W.-S. Chou, J. Gomez-Calderon and G. L. Mullen, Value sets of Dickson polynomials over finite fields, J. Number Theory 30 (1988), 334–344.

[4] M. Freedman, Priviate communication.

[5] G. H. Hardy and E. M. Wright, The Theory of Number, Oxford University Press, Oxford, UK, 1971.

[6] X. Hou, G. L. Mullen, J. A. Sellers, J. L. Yucas, Reversed Dickson polyno-mials over finite fields, Finite Fields Appl. 15 (2009), 748-773.

[7] R. Lidl, G.L. Mullen and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 65, Longman Group UK Limited 1993.

[8] R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. Vol. 20, Addison-Wesley, Reading, 1983.

### 56

[9] H. Meyn, On the construction of irreducible self-reciprocal polynomials over finite fields, Applicable Algebra in Engineering, Communication and Computing, 1 (1990), 43-53.

[10] The Online Encyclopedia of Integer Sequences, A001122, A093179, http://oeis.org/

[11] D. Wiedemann, An iterated quadratic extension of GF(2), Fibonacci Quart.

26 (1988), 290-295.

[12] http://www.fermatsearch.org/factors/composite.php

### Department of Mathematics National Central University E-mail: whyu@math.ncu.edu.tw

In this talk, I will talk about the history and background knowledge of equiangular line problems. Then, I will talk our contribution in this area. This talk is based on the joint work with Dr. Yen-Chi Lin.

### Tamkang University E-mail: hchsu0222@gmail.com

In this talk we will introduce the signed q-counting over partitions whose Ferrers diagrams fit inside a given partition, where the sign is the parity of the size and the enumerator statistic is the length. We will introduce several q-identities, exhibiting a certain pattern which we called the folding phenomenon.

Keywords: partition, Ferrers diagram, q-analogue, folding phenomenon

### References

[1] R.M. Adin, Y. Roichman, Equidistribution and sign-balance on 321-avoiding permutations, Sémin. Loth. Combin. 51 (2004) B51d.

[2] W.Y.C. Chen, L.W. Shapiro, L.L.M. Yang, Parity reversing involution on plane trees and 2-Motzkin paths, European J. Combin. 27 (2006) 283–289.

[3] J. Désarménien, D. Foata, The signed Eulerian numbers, Discrete Math.

99 (1992) 49–58.

[4] S.-P. Eu, S.-C. Liu, Y.-N. Yeh, Odd or even on plane trees, Discrete Math.

281 (2004) 189–196.

[5] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Sign-balance identities of Adin-Roichman type on 321-avoiding alternating permutations, Discrete Math.

312 (2012) 2228–2237.

[6] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Baxter Permutations, Maj-balances, and Positive Braids, Electronic J. Combin. 19 Issue 3 (2012) P26.

[7] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Two refined major-balance iden-tities on 321-avoiding involutions European J. Combin. 49 (2015) 250–264.

[8] S.-P. Eu, T.-S. Fu, Y.-J. Pan, P.-L. Yan, More on double Simsun permuta-tions, non-published manuscript.

[9] T. Mansour, Equidistribution and sign-balance on 132-avoiding permuta-tions, Sémin. Loth. Combin. 51 (2004) B51e.

### 59

[10] A. Reifegerste, Refined sign-balance on 321-avoiding permutations, Euro-pean J. Combin. 26 (2005) 1009–1018.

[11] A. Robertson, D. Saracino, D. Zeilberger, Refined restricted permutations, Ann. Comb. 6 (2002), 427–444.

[12] M. Shattuck, Parity theorems for statistics on permutations and Catalan words, Integers: Electronic J. Combin. Number Theory 5 (2005) #A07.

[13] R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin.

6 (1985) 383–406.

[14] R. Stanley. Some remarks on sign-balanced and maj-balanced posets. Adv.

Appl. Math. 34(4) (2005) 880–902.

[15] C.-T. Ting, Folding phenomena of some classes of permutations, Thesis, 2017.

[16] M. Wachs, An involution for signed Eulerian numbers, Discrete Math. 99 (1992) 59–62.

### 地 點 ： M 2 1 2 數 學 館

TMS Annual Meeting

## 2018 數 學 年 會

Speech Abstracts

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

### 吳金典

Chin-Tien Wu Mathematics in 3D imaging and its applications

1 3 : 3 0 - 1 3 : 5 5

### 黃杰森

Chieh-Sen Huang

Von Neumann stable, implicit finite volume WENO schemes for hyperbolic conservation laws

1 4 : 0 0 - 1 4 : 2 5

### 嚴健彰

Chien-Chang Yen

Self-Gravitational Force Calculation Using A Direct Method for Adaptive Mesh Refinement

1 4 : 3 0 - 1 4 : 5 5

### 游承書

Cheng-Shu You

A finite difference scheme for strongly coupled systems of singularly perturbed equations

1 5 : 2 0 - 1 5 : 4 5

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

### 吳宗信

Jong-Shinn Wu

RAPIT (Rigorous Advanced Plasma Integration Testbed):

A Parallel Scientific Computational Platform

### 李勇達

Yung-Ta Li A pseudospectral method for the solution of the Helmholtz equation

### 許佳璵

Chia-Yu Hsu The Study of Schooling Pattern of Lampreys ㏔㐠45&2'(熂漯㔛孬偖㩪條㠶䝜

### E-mail: ctw@math.nctu.edu.tw

In this talk, I shall introduce some basics mathematics in 3D imaging and its applications. Principles in 3D scanning, mesh generation from point cloud, mesh processing, texture mapping and color tuning, etc. will be presented. I would like to share my recent works in this area with audience, especially those challenges and difficulties that I am still struggling with.

### E-mail: csyou@fcu.edu.tw

We present a new approach to defining implicit WENO (iWENO) schemes for systems of hyperbolic conservation laws. The approach leads to schemes that are simple to implement, high order accurate, maintain local mass conservation, apply to general computational meshes, and appear to be fairly robust. We present third and fifth order finite volume schemes in one and two space dimen-sions. We show that these iWENO schemes are unconditionally stable in the sense of von Neumann stability analysis, assuming the solution is smooth. The solution is approximated efficiently by two or three degrees of freedom per com-putational mesh element, independent of the spatial dimension. In space, the degrees of freedom are reconstructed implicitly to give high order approximation, while avoiding shocks and steep fronts due to the WENO framework. In time, high order quadrature is employed to produce a one step scheme. The approach is quite general, and we apply it to advection-diffusion-reaction equations with simple diffusion a nd r eaction t erms. N umerical r esults o n n onuniform meshes in one and two space dimensions are presented. These explore the properties of the new schemes for solving hyperbolic conservation laws, advection-diffusion equations, advection-reaction equations, and the Euler system.

### Fu-Jen Catholic University E-mail: yen@math.fju.edu.tw

A direct approach for self-gravitational force calculation of second order accuracy based on uniform grid discretization has been proposed by Yen et al. The method improves the N-body calculation on accuracy using the exact integration of kernel functions and employs fast Fourier transform (FFT) to reduce the computational complexity to nearly linear. This direct approach is also free artificial boundary conditions. However, the uniform discretization is a limitation. Due to computational facility or power has been improved during the past decade, it promotes us to investigate the direct method for non-uniform grid discretization preserving second order accuracy and simulations in reality with the help of graphic process units (GPU) to speed up computational time. The proposed method is more flexible on grid discretization and has the potential to be applied to studies the gaseous morphology of disk galaxies and the planetary migration. This is a join work with Yao-Huan Tseng and Hsien Shang.

### E-mail: csyou@fcu.edu.tw

In this talk, we will consider the strongly coupled systems of singularly per-turbed convection-diffusion equations, where strong coupling means that the so-lution components in the system are coupled together through their first deriva-tives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we fist construct a so-called Il’in-Allen-Southwell (IAS) scheme for 1D systems and then extend the scheme to 2D systems by employing an al-ternating direction technique. From the numerical results, we can observe that when the perturbation parameter ε is small enough, the developed IAS scheme is fist order convergent in the discrete maximum norm uniformly in ε on uniform meshes. This is a joint work with Po-Wen Hsieh and Suh-Yuh Yang.

Keywords: boundary and interior layers, Il’in-Allen-Southwell scheme, sin-gularly perturbed convection-diffusion e quation, s trongly c oupled s ystem, uni-form convergence.

### Department of Mechanical Engineering National Chiao Tung University E-mail: chongsin@faculty.nctu.edu.tw

Many important and challenging science and engineering problems require modeling of com-plex plasma and flow physics applying hybridization of different continuum- and/or particle-based solvers. Examples may include plume analysis of reaction control thrusters on upper-stage rocket and satellite in orbit, rocket plume analysis at high altitude, aerodynamic analysis of atmospheric-pressure dielectric barrier discharge (DBD) actuator, radical distribution of atmospheric-atmospheric-pressure plasma jet, ion thruster plume analysis, and plasma distribution in etching and thin-film depo-sition chambers at low pressure, to name a few. These studies often utilize independent solvers developed previously and integrate them in a non-self-consistent approach, which makes their applications and future extension highly inflexible. Thus, a highly flexible simulation platform, which allows straightforward addition and integration of different solvers with a self-consistent approach while maintaining efficient computations, is strongly needed to tackle some challenging problems with complex plasma/flow physics.

In this talk, I will report our recent development of a new C++ object-oriented multi-physics simulation platform named Rigorous Advanced Plasma Integration Testbed (RAPIT) using unstructured-grid finite-volume method with parallel computing through MPI (message passing interface) on distributed-memory PC clusters. The proposed RAPIT with both embedded PDE and particle solver related objects can easily accommodate continuum- and/or particle-based solvers with some proper hybridization algorithm in a self-consistent way. For the former, it may include, but not limited to, the Navier-Stokes (NS) equation solver for general gas flow modeling and the plasma fluid modeling code for general low-temperature plasma modeling. For the latter, it may include the particle-in-cell Monte Carlo collision (PIC-MCC) solver for very low-pressure gas discharge simulation and the direct simulation Monte Carlo (DSMC) solver for rarefied neu-tral gas flow modeling. Many distinct features of RAPIT include single or multiple mesh(es) for different solvers or species with automatic interpolation relation, essentially the same source code for 2D and 3D problems due to nearly operator-like programming style, and embedded parallel implementation, among others. Some preliminary results of DSMC, PIC-MCC and NS equa-tion and fluid modeling solvers in many practical engineering problems will be presented in this talk. In addition, a byproduct of RAPIT, ultraMPP (ultra-fast Massively Parallel Processing), which is a parallel computing platform for PDE related solvers, will also be briefly introduced. It is designed to greatly reduce the development time of parallel 2D/3D codes from years to weeks while maintaining a highly manageable and consistent source coding framework for researchers.

Some major findings along with outlook are summarized at the end of this presentation.

Remark: Co-authors

Y. M.∼Lee and M.∼H. Hu Plasma Taiwan Innovative Corp.

Juh-bei City, Hsunchu County, Taiwan

### Department of Mathematics Fu-Jen Catholic University E-mail: ytli@math.fju.edu.tw

In this talk, we present a pseudospectral method for the Helmholtz equa-tion. The key of the numerical algorithm is to choose a suitable basis associated with the Legendre polynomials that has the following two features: (1) bound-ary conditions are met and (2) the linear system arising from discretizing the Helmholtz equation under the basis is easily solved. To interpretate the proce-dure of constructing such a basis, we first introduce two matrix decompositions which are the discrete analogues of the recursion formula and the orthogonal property of the Legendre polynomials, respectively. Subsequently the basis can be constructed through performing row/column operations on the matrix de-compositions. Numerical experiments are presented to validate the proposed method.

Keywords: Helmholtz equation; pseudospectral method; Legendre polyno-mials; tridiagonal matrix.

### E-mail: cyuhsu@fcu.edu.tw

The schooling pattern [1] in marine ecology is a common migration pattern for fishes of different swimming styles, such as carangiform of makrells, sub-carangiform of salmonids or anguiliform of eels [2]. This pattern is not only to move for food or survival, but also to avoid the predators and save the body energy loss, such as diadromous fishes of eels or salmons. In this talk, a model of multiple annugiliform swimmers, such as lamprey [3], is created to simulate the schooling pattern. The adaptive mesh refinement immersed boundary method is used for the numerical solution for the simulations. Moreover, the factors of body [4], such as body stiffness, spacing or body waveform, for swimming in schooling pattern will be investigated.

Keywords: lamprey, schooling pattern, adaptive mesh refinement immersed boundary method

### References

[1] A.D. Becker, H. Masoud, J. W. Newbolt1, M. Shelley, L. Ristroph1 Hydro-dynamic schooling of apping swimmers Natural Communication (2015), 1-8

[2] Eric D. Tytell The hydrodynamics of eel swimming, II. Effect of swimming speed J. of Exp. Biol., 207 (2004), 3265-3279.

[3] F. W. H. Beamish Swimming Performance of Adult Sea Lamprey, Petromy zon marinus, in Relation, to Veight and Temperature Trans. Amer. Fish.

Soc., (1974), NO.2, 355-358

[4] E. D. Tytell, M. C. Leftwich, C-Y Hsu, B. E. Griffth, A. H. Cohen, A.J.

Smits and C. Hamlet and L. J. Fauci Role of body stiffness in undulatory swimming: Insights from robotic and computational models Phys. Rev.

Flu., 1, (2016), 073202

### 機率 Probability 地 點 ： B 1 0 3 理 學 院

TMS Annual Meeting

## 2018 數 學 年 會

Speech Abstracts

ѱ䗜௤փ ঊ䗜௤փ

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

### 䁧丼ਿ

6KXQ&KL+VX 6RPH6WRFKDVWLF&RQWURO3UREOHPVLQWKH6WXG\RI)LQDQFH

1 3 : 3 0 - 1 3 : 5 5

### 举р㤇

6KDQJ<XDQ6KLX 2QWKH6WRFKDVWLF+HDW(TXDWLRQV

1 4 : 0 0 - 1 4 : 2 5

### ⍠㣭╠

-\\,+RQJ 6RPH/LPLW'LVWULEXWLRQVRI'LVFRXQWHG%UDQFKLQJ5DQGRP:DONV

1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

Outline